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Concavity & Inflection Points
AP Calculus AB · Unit 5 Analytical Applications
Learning Objectives
Determine concavity from the sign of f″(x)
Find inflection points where f″(x) changes sign
See f, f′, f″ simultaneously
Connect concavity to curve shape (cup up/down)
Key Concepts
f″(x) > 0:
Concave up (holds water) ∪
f″(x) < 0:
Concave down ∩
Inflection point:
f″ changes sign
f″ = 0 is necessary but NOT sufficient
Tags
concavity
inflection
second derivative
f″
FUNCTION
f(x) = x³ − 3x
f(x) = x⁴ − 6x² + 4
f(x) = sin(x)
f(x) = xe⁻ˣ
Cursor x
0.0
f(x)
0
f′(x)
0
f″(x)
0
Concavity
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